Almost Complex Structures

Almost Complex Structures on Symplectic Manifolds

Almost Complex Structure

An almost complex structure on a smooth manifold $M$ is a smooth bundle map $J:TM\to TM$ such that $J^2=-\text{id}$. The pair $(M,J)$ is called an almost complex manifold.

Compatible Almost Complex Structures

Let $(M,\omega )$ be a symplectic manifold. An almost complex structure $J$ on $M$ is said to be compatible (with $\omega$, or $\omega$-compatible) if $\omega (Ju,Jv)=\omega (u,v)$ and $\omega (u,Ju)>0$ for all non-zero $u\in TM$. That is $g(u,v):=\omega (u,Jv)$ admits a Riemannian metric on $M$.